Theorem reseq1d 5064

Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Hypothesis

Ref	Expression
reseqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
reseq1d	|- ( ph -> ( A |` C ) = ( B |` C ) )

Proof of Theorem reseq1d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 |- ( ph -> A = B )
2		reseq1 5059	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A |` C ) = ( B |` C ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   |` cres 4778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-res 4788

This theorem is used by:  reseq12d  5066  fun2ssres  5424  funcnvres2  5459  funimaexg  5468  fresin  5571  offres  6372  tfrlemisucaccv  6601  tfrlemi1  6608  tfr1onlemsucaccv  6617  tfrcllemsucaccv  6630  freceq1  6668  freceq2  6669  fseq1p1m1  10517  setsresg  13447  setscom  13449  gsumsplitfi  14255  gsum2dfilem2  14271  znle2  15095  dvcoapbr  15950  bj-charfundcALT  17057